Advertisements
Advertisements
Question
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Advertisements
Solution
Given that: a = cosθ + isinθ
∴ `(1 + a)/(1 - a) = (1 + cos theta + i sin theta)/(1 - cos theta - i sin theta)`
= `(1 + cos theta + i sin theta)/(1 - cos theta - i sin theta) xx (1 - cos theta + i sin theta)/(1 - cos theta + i sin theta)`
= `(1 - cos theta + i sin theta + cos theta - cos^2 theta + i sin theta cos theta + i sin theta - i sin theta cos theta + i^2 sin^2 theta)/((1 - cos theta)^2 - i^2 sin^2 theta)`
= `(1 + i sin theta - cos^2 theta + i sin theta - sin^2 theta)/(1 + cos^2 theta - 2 cos theta + sin^2 theta)`
= `(sin^2 theta + 2i sin theta - sin^2 theta)/(1 + 1 - 2 cos theta)`
= `(2i sin theta)/(2 - 2 cos theta)`
= `(2i sin theta)/(2(1 - cos theta))`
= `(i sin theta)/(1 - cos theta)`
= `(2 sin theta/2 cos theta/2.i)/(2sin^2 theta/2)`
= `cot theta/2 . i`
Hence, `(1 + a)/(1 - a) = icot theta/2`.
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Write (i25)3 in polar form.
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The principal value of the amplitude of (1 + i) is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
Find a and b if `1/("a" + "ib")` = 3 – 2i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(2 + 3i)(2 – 3i)
Evaluate the following : i93
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
The order relation is defined on the set of complex numbers.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
Show that `(-1 + sqrt3 "i")^3` is a real number.
